# FUZZY REGRESSION MODEL WITH MONOTONIC RESPONSE FUNCTION

• Choi, Seung Hoe (School of Liberal Arts and Science Korea Aerospace University) ;
• Jung, Hye-Young (Faculty of Liberal Education Seoul National University) ;
• Lee, Woo-Joo (Department of Mathematics Yonsei University) ;
• Yoon, Jin Hee (School of Mathematics and Statistics Sejong University)
• Accepted : 2018.05.18
• Published : 2018.07.31
• 189 10

#### Abstract

Fuzzy linear regression model has been widely studied with many successful applications but there have been only a few studies on the fuzzy regression model with monotonic response function as a generalization of the linear response function. In this paper, we propose the fuzzy regression model with the monotonic response function and the algorithm to construct the proposed model by using ${\alpha}-level$ set of fuzzy number and the resolution identity theorem. To estimate parameters of the proposed model, the least squares (LS) method and the least absolute deviation (LAD) method have been used in this paper. In addition, to evaluate the performance of the proposed model, two performance measures of goodness of fit are introduced. The numerical examples indicate that the fuzzy regression model with the monotonic response function is preferable to the fuzzy linear regression model when the fuzzy data represent the non-linear pattern.

#### Keywords

fuzzy regression model;monotonic response function;resolution identity theorem;LS method;LAD method

#### Acknowledgement

Supported by : National Research Foundation of Korea (NRF)

#### References

1. J. J. Buckley and T. Feuring, Linear and non-linear fuzzy regression: evolutionary algorithm solutions, Fuzzy Sets and Systems 112 (2000), no. 3, 381-394. https://doi.org/10.1016/S0165-0114(98)00154-7
2. C. W. Chen, C. H. Tsai, K. Yeh, and C. Y. Chen, S-curve regression of fuzzy method and statistical application, Proceeding of the 10th IASTED International Conference Artiﬁcial Intelligence and Soft Computing (2006), 215-220.
3. S. H. Choi and J. J. Buckley, Fuzzy regression using least absolute deviation estimators, Soft Computing 12 (2008), 257-263.
4. S. H. Choi, H. Y. Jung, W. J. Lee, and J. H Yoon, On Theil's method in fuzzy linear regression models, Commun. Korean Math. Soc. 31 (2016), no. 1, 185-198. https://doi.org/10.4134/CKMS.2016.31.1.185
5. S. H. Choi and J. H. Yoon, General fuzzy regression using least squares method, International J. Systems Science 41 (2010), 477-485. https://doi.org/10.1080/00207720902774813
6. P. Diamond, Fuzzy least squares, Inform. Sci. 46 (1988), no. 3, 141-157. https://doi.org/10.1016/0020-0255(88)90047-3
7. N. R. Draper and H. Smith, Applied Regression Analysis, John Wiley & Sons, Inc., New York, 1966.
8. D. G. Hong and C. H. Hwang, Fuzzy nonlinear regression model based on LS-SVM in feature space, In: International Conference on Fuzzy Systems and Knowledge Discovery, 208-216, Springer, Berlin, Heidelberg, 2006.
9. L. Hu, R. Wu, and S. Shao, Analysis of dynamical systems whose inputs are fuzzy stochastic processes, Fuzzy Sets and Systems 129 (2002), no. 1, 111-118. https://doi.org/10.1016/S0165-0114(01)00073-2
10. R. I. Jennrich, An Introduction to Computational Statistics -Regression Analysis, Englewood Cliﬀs, NJ: Prentice-Hall International Inc., 1995.
11. H.-Y. Jung, J. H. Yoon, and S. H. Choi, Fuzzy linear regression using rank transform method, Fuzzy Sets and Systems 274 (2015), 97-108. https://doi.org/10.1016/j.fss.2014.11.004
12. B. Kim and R. R. Bishu, Evaluation of fuzzy linear regression models by comparison membership function, Fuzzy Sets and Systems 100 (1998), 343-352. https://doi.org/10.1016/S0165-0114(97)00100-0
13. H. K. Kim, J. H. Yoon, and Y. Li, Asymptotic properties of least squares estimation with fuzzy observations, Inform. Sci. 178 (2008), no. 2, 439-451. https://doi.org/10.1016/j.ins.2007.07.010
14. I. K. Kim, W. J. Lee, J. H. Yoon, and S. H. Choi, Fuzzy regression model using trapezoidal fuzzy numbers for re-auction data, International Journal of Fuzzy Logic and Intelligent Systems 16 (2016), 72-80. https://doi.org/10.5391/IJFIS.2016.16.1.72
15. W. J. Lee, H. Y. Jung, J. H. Yoon, and S. H. Choi, The statistical inferences of fuzzy regression based on bootstrap techniques, Soft Computing 19 (2015), 883-890. https://doi.org/10.1007/s00500-014-1415-5
16. M. Ming, M. Friedman, and A. Kandel, General fuzzy least squares, Fuzzy Sets and Systems 88 (1997), no. 1, 107-118. https://doi.org/10.1016/S0165-0114(96)00051-6
17. S. M. Taheri and M. Kelkinnama, Fuzzy linear regression based on least absolutes deviations, Iran. J. Fuzzy Syst. 9 (2012), no. 1, 121-140, 169.
18. H. Tanaka, I. Hayashi, and J. Watada, Possibilistic linear regression analysis for fuzzy data, European J. Oper. Res. 40 (1989), no. 3, 389-396. https://doi.org/10.1016/0377-2217(89)90431-1
19. H. Tanaka, S. Uejima, and K. Asai, Linear regression analysis with fuzzy model, IEEE Trans. System Man Cybernetic 12 (1982), 903-907. https://doi.org/10.1109/TSMC.1982.4308925
20. H.-C. Wu, Analysis of variance for fuzzy data, Internat. J. Systems Sci. 38 (2007), no. 3, 235-246. https://doi.org/10.1080/00207720601157997
21. R. Xu, S-curve regression model in fuzzy environment, Fuzzy Sets and Systems 90 (1997), no. 3, 317-326. https://doi.org/10.1016/S0165-0114(96)00120-0
22. L. A. Zadeh, Fuzzy sets, Information and Control 8 (1965), 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X
23. L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning. I, Information Sci. 8 (1975), 199-249. https://doi.org/10.1016/0020-0255(75)90036-5
24. X. Zhang, S. Omac, and H. Aso, Fuzzy regression analysis using RFLN and its application, Proc. FUZZ-IEEE'97 1 (1997), 51-56.